© 2021 Jabar Mahmood et al.The advancements in Vehicular Ad Hoc Networks (VANETs) require more intelligent security protocols that ultimately provide unbreakable security to vehicles and other components of VANETs. VANETs face various types of security pitfalls due to the openness characteristics of the VANET communication infrastructure. Researchers have recently proposed different mutual authentication schemes that address security and privacy issues in vehicle-to-vehicle (V2V) communication. However, some V2V security schemes suffer from inadequate design and are hard to implement practically. In addition, some schemes face vehicle traceability and lack anonymity. Hence, this paper's primary goal is to enhance privacy preservation through mutual authentication and to achieve better security and performance. Therefore, this article first describes the vulnerabilities of a very recent authentication scheme presented by Vasudev et al. Our analysis proves that the design of Vasudev et al.'s scheme is incorrect, and resultantly, the scheme does not provide mutual authentication between a vehicle and vehicle server when multiple vehicles are registered with the vehicle sever. Furthermore, this paper proposes a secure message transmission scheme for V2V in VANETs. The proposed scheme fulfills the security and performance requirements of VANETs. The security analysis of the proposed scheme using formal BAN and informal discussion on security features confirm that the proposed scheme fulfills the security requirements, and the performance comparisons show that the proposed scheme copes with the lightweightness requirements of VANETs.